# Full text of "Analytical Mechanics"

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CENTER OF MASS AND MOMENT OF INERTIA 1 (b) Suppose the square OA to represent a plate of positive mass a the square O'A to represent a plate of negative mass. Then if the t plates have the same thickness and density the positive and the negat: masses annul each other in the square O'A. Therefore the two squ; plates form a system which is equivalent to the actual plate represeni by the shaded area of the figure. Hence the center of mass of the squ; plates is also the center of mass of the given plate. The masses of the square plates are <r&2 and nates of their centers of mass are era2, while the coor -, i = y'=~ and 2b — a Therefore the coordinates of the center of mass of the two are T o & i / o\ 2 & — a = b*-\~ab — a2 2 (a + 6) ' which are identical with those obtained by the first method. PROBLEMS. 1. Find the center of mass of the homogeneous plates indicated by following figures: (e) Find 2. A sphere of radius 6 has a spherical cavity of radius a. center of mass if the distance between the centers is c. 3. A right cone is cut from a right circular cylinder of the same b and altitude. Find the center of mass of the remaining solid. 4. A right cone is cut from a hemisphere of the same base and altitu Find the center of mass of the remaining solid. 6. A right circular cone is cut from another right circular cone of same base but of greater altitude. Find the center of mass of the rems ing solid. 6. A right circular cone is cut from the paraboloid of revolution gei